Making Equilateral Triangles Outdoors

On Friday morning, our class bundled up and headed out to the outdoor classroom for math.  Cat was with us that morning as we continued our investigation of geometry and measurement.  The first challenge was to break into groups of three and then create an equilateral triangle using only yarn, scissors and their bodies.  A few students asked for tape measures, but the challenge was to construct the shape without a standard measurement tool.  

So why take this outdoors?

This is certainly an activity that could be done indoors, but this group has demonstrated that the level of ingenuity, problem-solving and cooperation is greater in our outdoor classroom.

So how did they solve the problem?

I noticed one group that started with a length of yarn.  The students positioned themselves in a triangular shape, estimating that they were an equal distance from each other.  I asked, “How do you know it’s an equilateral triangle?”  They responded, “We know because we’re standing the same distance from each other.  I pushed a little harder, “But how do you know for sure that the sides are of equal length?”  They then folded the yarn into thirds but quickly saw that their lengths were not quite equal.  A simple adjustment guaranteed that the sides were now congruent.

 Another group insisted that the only way to build an equilateral triangle was to measure using a standard tool.  Because the logs of the community space had already been measured at 15 feet each, the yarn triangle they created was an exact copy of the logs.  T said, “We already know this one is equilateral.”  They used their bodies as vertices, pulling the yarn behind their backs at the waistline.  Cat noticed that this distorted the shape somewhat, and C said that they could turn “sideways”, making it “thinner” and therefore closer to a point.  He also said that if they used their fingers, it would be “even better”.  What if we had not gone outdoors?  Here, they were able to build on prior experience to guide them in their cooperative problem-solving.

Cat asked them if it would be possible to make an even smaller equilateral triangle, and they struggled to come up with a strategy.  N came over and told them about how her group had accomplished it.  They had cut off a “random length” of yarn and then used that piece to measure off two identical pieces.  

By not allowing the students to use a standard measurement tool, the students got to the nature of congruence (equal measure), of measurement itself, and of proving something definitively.  Even the students said that they thought it would be “much easier” than it was.  After all, they could name and describe an equilateral triangle.  But building one was an entirely different matter. Years from now, in high school geometry, they will have to construct an equilateral triangle –again without using measurement tools– and prove how they know the sides are congruent.